Introduction to grouping | Algebra I | Khan Academy

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Analyze t ^ 2 + 8t + 15 So let’s think about what happens if we hit two binomial terms (t + a) (t + b) And I use t here because t is the variable in the polynomial What we will need to analyze, if you hit them Using the distribution feature twice or using FOIL You will get t x t which equals t ^ 2 + t x b That is, bt, + a x t and this equals at + a x b Ie ab, we have hit every phrase here With every phrase there, we have two phrases, two t phrases We can call them this. This bt + at So we can combine them, and we get t ^ 2 + a + bt, I can write that this is b + at as well + ab, if we compare this to this here We will see that we have similar pattern Coefficient, the second degree ferry coefficient is 1 The coefficient of the second degree expression here is 1 We don’t have to write it, then a + b is the coefficient of the expression t And 8 here, these 8 can be a + b Finally, the constant ab is 15 This is going to be 15. So if we want to extract this as a common factor We will have to find values \u200b\u200bfor a and b times their product 15 And a total of 8. In general, in general if you have seen – I’m going to write it more traditionally using the variable x– If you see something in the form of x ^ 2 + bx + c The coefficient here is 1, so you will have to find Two numbers together equal to this And their product is equal to this That is, their sum is equal to 8, and their product is equal to 15 What are the two numbers that sum 8 and Whose product is 15? If we extract the 15 We have 1 and 15, their sum is not equal to 8 3 and 5, these two are equal to 8 So a and b are going to be 3 and 5, a and b This could be 3 x 5 and 8 words n 3 + 5 Now we can go straight and analyze these and say This is (t + 3), (t + 5), since we actually found the values \u200b\u200bof a and b But what I want to do is to analyze it by grouping So I will take a step back In terms of what I have shown you, then this is first, this polynomial I would type it as t ^ 2 +, instead of 8t I’m going to write it as a sum of at + bt Or as a sum of 3t + 5t, then 3t + 5t I started from here and will go to this step Where I split the middle phrase into parameters Sum of 8, then finally + 15 Now, I can analyze by grouping In these two expressions we have the common factor t These two have the common factor 5 So let’s extract the factor t from the first statement Or this part of the phrase, so t x (t + 3) + Here if we extract the 5, we get 5 X (t + 3), 5t ÷ 5 = t, 15 ÷ 5 = 3 Now we can extract the factor t + 3 We have t + 3 factorial of both of these terms So let us extract it, let us extract it It becomes (t + 3) (t + 5) × – I will write it more carefully– X (t + 5), and you’re done! In fact, we do not have to To do this assembly step, and yet I hope you see That this worked, we can say, see I have two numbers of this pattern here Their sum is 8 and their product is 15 So this (t + 3) (t + 5) or (t + 5) (t + 3), in any way you want

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